Starting with a sampled spectrum , , typically obtained from a DFT, we can interpolate by taking the DTFT of the IDFT which is not periodically extended, but instead zero-padded [264]:^{3.8}
(The aliased sinc function, , is derived in §3.1.) Thus, zero-padding in the time domain interpolates a spectrum consisting of samples around the unit circle by means of `` interpolation.'' This is ideal, time-limited interpolation in the frequency domain using the aliased sinc function as an interpolation kernel. We can almost rewrite the last line above as , but such an expression would normally be defined only for , where is some integer, since is discrete while is continuous.
Figure F.1 lists a matlab function for performing ideal spectral interpolation directly in the frequency domain. Such an approach is normally only used when non-uniform sampling of the frequency axis is needed. For uniform spectral upsampling, it is more typical to take an inverse FFT, zero pad, then a longer FFT, as discussed further in the next section.